Under mixing conditions for the process \(A(X(t))\) , for a sufficiently
large \(\tau _{b}\)  the random variable \(A_{\tau _{b}}\)  can be considered
by block averaging as a sum of independent and identically distributed
(iid) random variables, for which explicit results for the convergence
of estimators of large deviation functions have been obtained [1]}.
Euristically one expects that the integral in each block corresponds to a sum over \(\tau _b/\tau _c\)  values.
The \(N_b\)  values from the original time series are then taken as independent realizations of such sum of iid random variables.
This approach allows to study precisely the convergence of the
estimates of the large deviation functions. Note that the same approach
can be followed if instead of a single long simulation we have an
ensemble of shorter simulations each of length \(\tau _b\) .
